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Tie Knots as Random Walks

We represent knot sequences as random walks on a triangular lattice. The axes r,c,l correspond to the three regions R,C,L through which the active end can be wound and the unit vectors represent the corresponding moves (Fig. 3); we omit the directional notation and the terminal action T. Since all knot sequences end with and alternate between and , all knots of odd half-winding number h begin with while those of even h begin with . Our simplified random walk notation is thus unique, and we only make use of the directional notation in the context of move sequences.

The three-fold symmetry of the move regions implies that only steps along the positive lattice axes are acceptable and, as in the case of moves, no consecutive steps can be identical; the latter condition makes our walk a second order Markov, or persistent, random walk. Nonetheless, every site on the lattice can be reached since, e.g., and .

Figure 3: A tie knot may be represented by a persistent random walk on a triangular lattice, beginning with and ending with or . Only steps along the positive r,c and l axes are permitted and no two consecutive steps may be the same. Shown here is the Four-in-Hand, indicated by the walk .

The evolution equations for the persistent random walk appear as

where is the conditional probability that the walker is at point (r,c,l) on the nth step, having just taken a step along the positive r-axis, p is conditioned on a step along the positive c-axis, etc. The unconditional probability of occupation of a site, U, may be expressed .

Yong Mao
Sat Nov 7 16:03:57 GMT 1998